Method for calculation of natural frequency of multi-segment continuous beam

ABSTRACT

A displacement spring and a rotational spring are arranged on both ends of the multi-segment continuous beam to simulate arbitrary boundary conditions, and a lateral displacement function of the multi-segment continuous beam over a whole segment is constructed. A strain energy, an elastic potential energy of simulated springs at a boundary, a maximum value of a kinetic energy, and a Lagrangian function of the multi-segment continuous beam are calculated. The improved Fourier series of the displacement function is substituted into the Lagrange function. An extreme value of each undetermined coefficient in the improved Fourier series in the Lagrangian function is taken to obtain a system of homogeneous linear equations which is further converted into a matrix. An eigenvalue problem of the standard matrix is solved for to obtain the natural frequency.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority from Chinese PatentApplication No. 202010343754.2, filed on Apr. 27, 2020. The content ofthe aforementioned application, including any intervening amendmentsthereto, is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present application relates to beam structures, and moreparticularly to a method for calculation of a natural frequency of amulti-segment continuous beam.

BACKGROUND

Multi-segment continuous beams are defined as stepped rods with bendingas the main deformation, and multi-segment beam components are widelyapplied in engineering, such as stepped shafts for supporting rotatingparts and transmitting motion and power in power machinery, steppeddrill strings and oil rods in oil drilling engineering, stepped pistonrods in engines, and workpieces in turning. The vibration of thecontinuous beams is a basic subject in mechanical vibration, and naturalfrequencies of the multi-segment continuous beams are affected bymultiple factors, such as cross-sectional shapes, lengths of segmentedrods, materials, and lengths of beams. The existing literatures haveprovided natural frequency equations of straight rods with a constantcross-section under conventional classical boundary conditions (such asclamped constrain, simply supported constrain, free boundary), so thatthe natural frequency value can be obtained by solving the correspondingequations. However, under a given boundary condition, in order todetermine the natural frequency of a multi-segment continuous beam, thecorresponding frequency equations are relatively complicated, and alarge amount of calculation is required. For the calculation of bendingvibration of two-segment stepped beams, there is no systematicderivation and calculation of the natural frequency of bending vibrationof stepped multi-segment beams in the existing literatures. At the sametime, the applicable formula for calculating the natural frequency ofthe bending of the stepped multi-segment beam under given elasticboundary conditions is not found. Therefore, it is necessary to providea method for calculation of a natural frequency of bending vibrations ofeach order of the multi-segment continuous beam.

SUMMARY

In order to solve the above-mentioned technical defects, the presentdisclosure provides a method for calculation of a natural frequency of amulti-segment continuous beam. The derivation and the calculation of thenatural frequency of the multi-segment continuous beam under an elasticboundary condition are performed, which can quickly obtain themulti-order natural frequencies of bending of a multi-segment beam,where multiple segments of the multi-segment beam have differentcross-sectional shapes, different materials and different lengths. Thus,the method of the present disclosure is easy to popularize and use.

To achieve the above-mentioned object, the present disclosure provides amethod for calculation of a natural frequency of a multi-segmentcontinuous beam, comprising:

(1) arranging a displacement spring and a rotational spring on each of afirst end and a right end of the multi-segment continuous beam tosimulate arbitrary boundary conditions;

(2) constructing a lateral displacement function of the multi-segmentcontinuous beam over a full length thereof, and expressing the lateraldisplacement function in a form of an improved Fourier series, whereinthe improved Fourier series is formed by adding four auxiliary functionsinto a classic Fourier series;

(3) calculating a strain energy of the multi-segment continuous beam;

(4) calculating an elastic potential energy of the displacement springand the rotational spring at a boundary of the multi-segment continuousbeam;

(5) calculating a maximum value of a kinetic energy of the multi-segmentcontinuous beam;

(6) calculating the Lagrangian function of the multi-segment continuousbeam;

(7) substituting the improved Fourier series of the lateral displacementfunction into the Lagrange function;

(8) taking an extreme value of each of undetermined coefficients in theimproved Fourier series in the Lagrangian function to let a partialderivative be zero, so as to obtain a system of homogeneous linearequations;

(9) converting the system of homogeneous linear equations obtained intoa matrix form; and

(10) solving for an eigenvalue problem of the matrix to obtain a naturalfrequency of each order of the multi-segment continuous beam.

In an embodiment, in step (1), a stiffness value of the displacementspring and a stiffness value of the rotational spring stiffness at oneboundary are respectively denoted as k₁ and K₁, and a stiffness value ofthe displacement spring and a stiffness value of the rotational springat the other boundary are respectively denoted as k₂ and K₂; when theboundary is a clamped boundary, the stiffness value of the displacementspring and the stiffness value of the rotational spring need to be setto infinity at the same time, and the stiffness value of thedisplacement spring and the stiffness value of the rotational spring areset to 10¹³, respectively; when the boundary is a free boundary, thestiffness value of the displacement spring and the stiffness value ofthe rotational spring are set to zero; when the boundary is a simplysupported boundary, the stiffness value of the displacement spring isset to 10¹³, and the stiffness value of the rotational spring is 0; andwhen the stiffness value of the displacement spring and the stiffnessvalue of the rotational spring are finite values, an elastic constraintboundary condition is simulated.

In an embodiment, the lateral displacement function of the multi-segmentcontinuous beam over a whole segment expressed in the form of theimproved Fourier series in step (2) is:

$\begin{matrix}{{{W(x)} = {{\sum\limits_{n = 0}^{9}{a_{n}{\cos\left( {\lambda_{n}x} \right)}}} + {\sum\limits_{n = {- 4}}^{- 1}{a_{n}{\sin\left( {\lambda_{n}x} \right)}}}}};} & (1)\end{matrix}$

wherein x ∈[0,L]; a_(n) is an undetermined constant; and λ_(n)=nπ/L

In an embodiment, the strain energy of the multi-segment continuous beamstructure in step (3) is:

$\begin{matrix}{{V_{P} = {{\frac{1}{2}E_{1}I_{1}{\int_{0}^{L_{1}}{\left( \frac{d^{2}w}{{dx}^{2}} \right)^{2}{dx}}}} + {\frac{1}{2}{\sum\limits_{i = 2}^{i = p}{E_{i}I_{i}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\left( \frac{d^{2}w}{dx^{2}} \right)^{2}{dx}}}}}}}};} & (2)\end{matrix}$

wherein a total length of the multi-segment continuous beam is L; themulti-segment continuous beam is divided into p segments; a length of ani-th segment is L_(i); Vp is the strain energy of the multi-segmentcontinuous beam under arbitrary boundary conditions; E_(i) is an elasticmodulus of the i-th segment, and I_(i) is a moment of inertia of a crosssection of the i-th segment.

In an embodiment, the elastic potential energy Vs of the displacementspring and the rotational spring at the boundary of the multi-segmentcontinuous beam in step (4) is:

$\begin{matrix}{V_{s} = {\frac{1}{2}{\left( \left. {k_{1}w^{2}} \middle| {}_{x = 0}{+ {K_{1}\left( \frac{\partial w}{\partial x} \right)}^{2}} \middle| {}_{x = 0}{{+ k_{2}}w^{2}} \middle| {}_{x = L}{+ {K_{2}\left( \frac{\partial w}{\partial x} \right)}^{2}} \right|_{x = L} \right).}}} & (3)\end{matrix}$

In an embodiment, a form of a modal solution of the multi-segmentcontinuous beam is assumed based on a variable separation method in step(2) as:

w(x,t)=W(x)e ^(iwt)   (4);

wherein i is an imaginary unit, and w is the natural frequency of themulti-segment continuous beam.

In an embodiment, the maximum value of the kinetic energy of themulti-segment continuous beam in step (5) is:

$\begin{matrix}{{T_{\max} = {{\frac{1}{2}{\rho(x)}{\int_{0}^{L}{{S(x)}\left( \frac{dw}{dt} \right)^{2}dx}}} = {\frac{\omega^{2}}{2}{\int_{0}^{L}{{\rho(x)}{S(x)}w^{2}dx}}}}}.} & (5)\end{matrix}$

In an embodiment, the Lagrangian function of the multi-segmentcontinuous beam in step (6) is:

$\begin{matrix}{L = {{V_{\max} - T_{\max}} = {V_{p} + V_{s} - {T_{\max}.}}}} & (6)\end{matrix}$

In an embodiment, in step (8), the partial derivative of the underminedcoefficient a_(n) (n=−4, −3, . . . , 9) is calculated item by item inthe Lagrangian function, to obtain the system of homogeneous linearequations:

[(M ₁ + . . . +M _(p))ω²−(Kp ₁ + . . . +Kp _(p) +Ks ₁ +Ks ₂ +Ks₃ +Ks₄)]A=0   (7);

wherein A={a⁻⁴, a⁻³, . . . , a₈, a₉}^(T),

${{Kp}_{1} = {E_{1}{I_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}},\ldots$ ${Kp_{p}} = {E_{p}{I_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}$ ${Kp_{p}} = {E_{p}{I_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}$ ${M_{1} = {\rho_{1}{A_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{f_{1}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{1}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{1}f_{m}{dx}}} \\{\int_{0}^{L_{1}}{f_{2}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{2}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{2}f_{m}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{f_{m}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{m}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{m}f_{m}{dx}}}\end{bmatrix}}}},\ldots$ $M_{p} = {\rho_{p}{{A_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{1}dx}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{m}dx}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{1}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{m}dx}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{1}dx}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{m}dx}}\end{bmatrix}}.}}$

In an embodiment, a condition for the system of the homogeneous linearequations to have a nontrivial solution in the step (8) is: a value ofcoefficient determinant of the system of the homogeneous linearequations is zero, to obtain a frequency equation.

Compared to the prior art, the present invention has followingbeneficial effects.

The method of the present invention can realize systematical derivationand calculation of the natural frequency of the multi-segment continuousbeam under an elastic boundary condition. Based on this method, thenatural frequencies of the multi-segment beam can be quickly obtainedwhen multiple segments of the multi-segment beam have differentcross-sectional shapes, different materials and different lengths.Therefore, the method of the present invention has broad applicationprospects.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure will be further described in detail inconjunction with the accompanying drawings.

The figure is a schematic diagram of a multi-segment continuous beammodel under arbitrary boundary conditions according to an embodiment ofthe present disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS

In order to make the technical means, inventive features, objectives andeffects of the present disclosure easy to understand, the presentdisclosure will be further illustrated below in conjunction withspecific embodiments.

As shown in the figure, the embodiment provides a method for calculationof a natural frequency of a multi-segment continuous beam, including thefollowing steps.

(1) A displacement spring and a rotational spring are arranged on eachof a left end and a right end of the multi-segment continuous beam tosimulate arbitrary boundary conditions.

(2) A lateral displacement function of the multi-segment continuous beamover a whole segment is constructed and consists of an undetermined modeshape function and an exponential function of an undetermined vibrationfrequency. The undetermined mode shape function is expressed in a formof an improved Fourier series, where the improved Fourier series isformed by adding four auxiliary functions into a classic Fourier series.

(3) A strain energy of the multi-segment continuous beam is calculated.The multi-segment continuous beam is a straight rod, such asBernoulli-Euler beams.

(4) An elastic potential energy of simulated springs at a boundary ofthe multi-segment continuous beam is calculated.

(5) A maximum value of a kinetic energy of the multi-segment continuousbeam is calculated.

(6) A Lagrangian function of the multi-segment continuous beam iscalculated.

(7) The improved Fourier series of the lateral displacement function issubstituted into the Lagrange function.

(8) An extreme value of each undetermined coefficient in the improvedFourier series in the Lagrangian function is taken to let a partialderivative be zero, so as to obtain a system of homogeneous linearequations.

(9) The system of homogeneous linear equations is converted into amatrix form.

(10) An eigenvalue problem of the standard matrix is solved for throughMathematica, to obtain a natural angular frequency of each order of themulti-segment continuous beam.

In step (1), a stiffness value of the displacement spring and astiffness value of the rotational spring at a left boundary arerespectively denoted as k1 and K₁, and a stiffness value of thedisplacement spring and a stiffness value of the rotational spring at aright boundary are respectively denoted as k₂ and K₂. When the boundaryis a clamped boundary, the stiffness value of the displacement springand the stiffness value of the rotational spring need to be set toinfinity at the same time, and the stiffness value of the displacementspring and the stiffness value of the rotational spring are set to 10¹³,respectively. When the boundary is a free boundary, the stiffness valueof the displacement spring and the stiffness value of the rotationalspring can be set to zero. When the boundary is a simply supportedboundary, the stiffness value of the displacement spring is set to 10¹³,and the stiffness value of the rotational spring is set to 0. When thestiffness value of the displacement spring and the stiffness value ofthe rotational spring are finite values, an elastic constraint boundarycondition can be simulated.

A form of a modal solution of the multi-segment continuous beam isassumed based on a variable separation method as:

w(x,t)=W(x)e ^(iwt)   (4);

where i is an imaginary unit; W(x) is the vibrational model function;and co is the natural frequency of the multi-segment continuous beam.

The vibrational model function W(x) is expressed in a form as follows:

$\begin{matrix}{{{W(x)} = {{\sum\limits_{n = 0}^{9}{a_{n}{\cos\left( {\lambda_{n}x} \right)}}} + {\sum\limits_{n = {- 4}}^{- 1}{a_{n}{\sin\left( {\lambda_{n}x} \right)}}}}};} & (1)\end{matrix}$

where x ∈[0,L]; a_(n)(n=−4, −3, . . . , 9) is an undetermined constant;and λ_(n)=nπ/L.

The strain energy of the multi-segment continuous beam consists ofstrain energies of segments of the multi-segment continuous beam and isexpressed as:

$\begin{matrix}{{V_{P} = {{\frac{1}{2}E_{1}I_{1}{\int_{0}^{L_{1}}{\left( \frac{d^{2}w}{{dx}^{2}} \right)^{2}{dx}}}} + {\frac{1}{2}{\sum\limits_{i = 2}^{i = p}{E_{i}I_{i}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\left( \frac{d^{2}w}{dx^{2}} \right)^{2}{dx}}}}}}}};} & (2)\end{matrix}$

the strain energy of each segment of the multi-segment continuous beamis expressed as:

${{V_{P1} = {\frac{1}{2}E_{1}I_{1}{\int_{0}^{L_{1}}{\left( \frac{d^{2}w}{{dx}^{2}} \right)^{2}dx}}}}{V_{P2} = {\frac{1}{2}E_{2}I_{2}{\int_{L_{1}}^{L_{1} + L_{2}}{\left( \frac{d^{2}w}{dx^{2}} \right)^{2}dx}}}}};$…

where a total length of the multi-segment continuous beam is L; themulti-segment continuous beam is divided into p segments; and a lengthof the i-th segment is Li; Vp is the strain energy of the multi-segmentcontinuous beam under arbitrary boundary conditions; Ei is an elasticmodulus of the i-th segment, and is a moment of inertia of of a crosssection of the i-th segment.

The elastic potential energy Vs of the simulated spring at the boundaryof the multi-segment continuous beam is:

$\begin{matrix}{V_{s} = {\frac{1}{2}{\left( \left. {k_{1}w^{2}} \middle| {}_{x = 0}{+ {K_{1}\left( \frac{\partial w}{\partial x} \right)}^{2}} \middle| {}_{x = 0}{{+ k_{2}}w^{2}} \middle| {}_{x = L}{+ {K_{2}\left( \frac{\partial w}{\partial x} \right)}^{2}} \right|_{x = L} \right).}}} & (3)\end{matrix}$

The maximum kinetic energy of the multi-segment continuous beam is:

$\begin{matrix}{{T_{\max} = {{\frac{1}{2}{\rho(x)}{\int_{0}^{L}{{S(x)}\left( \frac{dw}{dt} \right)^{2}dx}}} = {\frac{\omega^{2}}{2}{\int_{0}^{L}{{\rho(x)}{S(x)}w^{2}dx}}}}}.} & (5)\end{matrix}$

The Lagrangian function of the multi-segment continuous beam is:

L=V _(max) −T _(max=V) _(p) _(+V) _(s) _(−T) _(max)   (6).

The partial derivative of the undetermined coefficient an (n=−4, −3, . .. , 9) is calculated item by item in the Lagrangian function to obtainthe system of homogeneous linear equations:

[(M ₁ + . . . +M _(p))ω²(Kp ₁ + . . . +Kp _(p) +Ks ₁ +Ks ₂ +Ks ₃ +Ks₄)]A=0   (7);

where A={a⁻⁴, a⁻³, . . . , a₈, a₉}^(T),

${{Kp}_{1} = {E_{1}{I_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}},\ldots$ ${Kp_{p}} = {E_{p}{I_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}$ ${{Ks}_{1} = {k_{1}\begin{bmatrix}{f_{1}f_{1}} & {f_{1}f_{2}} & \ldots & {f_{1}f_{m}} \\{f_{1}f_{2}} & {f_{2}f_{2}} & \ldots & {f_{2}f_{m}} \\\vdots & \vdots & \vdots & \vdots \\{f_{1}f_{m}} & {f_{2}f_{m}} & \ldots & {f_{m}f_{n}}\end{bmatrix}}_{x = 0}},{{Ks}_{2} = {k_{2}\begin{bmatrix}{f_{1}f_{1}} & {f_{1}f_{2}} & \ldots & {f_{1}f_{m}} \\{f_{1}f_{2}} & {f_{2}f_{2}} & \ldots & {f_{2}f_{m}} \\\vdots & \vdots & \vdots & \vdots \\{f_{1}f_{m}} & {f_{2}f_{m}} & \ldots & {f_{m}f_{n}}\end{bmatrix}}_{x = L}},{{Ks}_{3} = {K_{1}\begin{bmatrix}{\frac{{df}_{1}}{dx}\frac{{df}_{1}}{dx}} & {\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} \\{\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} \\\vdots & \vdots & \vdots & \vdots \\{\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} & \ldots & {\frac{{df}_{m}}{dx}\frac{{df}_{m}}{dx}}\end{bmatrix}}_{x = 0}},{{Ks}_{4} = {{K_{2}\begin{bmatrix}{\frac{{df}_{1}}{dx}\frac{{df}_{1}}{dx}} & {\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} \\{\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} \\\vdots & \vdots & \vdots & \vdots \\{\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} & \ldots & {\frac{{df}_{m}}{dx}\frac{{df}_{m}}{dx}}\end{bmatrix}}_{x = L}{M_{1} = {\rho_{1}{A_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{f_{1}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{1}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{1}f_{m}{dx}}} \\{\int_{0}^{L_{1}}{f_{2}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{2}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{2}f_{m}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{f_{m}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{m}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{m}f_{m}{dx}}}\end{bmatrix}}}}}},{{\ldots M_{p}} = {\rho_{p}{{A_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{1}dx}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{m}dx}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{1}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{m}dx}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{1}dx}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{m}dx}}\end{bmatrix}}.}}}$

A condition for the system of the homogeneous linear equation to have anontrivial solution is: a value of the coefficient determinant of thesystem of the homogeneous linear equation is zero, to obtain a frequencyequation.

In the embodiment, through the above-mentioned calculation steps, astiffness matrix and a mass matrix of the multi-segment stepped beamwith the following common boundary conditions can be derived, and thenatural frequency values of the beam in different orders can be obtainedby solving for the eigenvalue problem. The boundary conditions include:

(1) one end of the beam is simply supported and hinged, and the otherend of the beam is clamped;

(2) one end of the beam is a simply supported and hinged, and the otherend of the beam is free;

(3) one end of the beam is clamped, and the other end is free;

(4) both ends of the beam are clamped;

(5) one end of the beam is simply supported and hinged, and the otherend of the beam is constrained by a wire spring and a torsion spring;

(6) one end of the beam is clamped, and the other end of the beam isconstrained by a wire spring and a torsion spring; and

(7) both ends of the beam are constrained by a wire spring and a torsionspring.

For a multi-segment continuous beam with any one of the above-mentionedboundary conditions, after obtaining the expressions of its mass matrixand stiffness matrix, its cross-sectional shape, cross-sectionaldimensions, total length and lengths of segments of the beam, andmaterial parameters of the segments of the beam can be changedarbitrarily, and the circular frequency values of each order of themulti-segment continuous beam under corresponding changes can be quicklyobtained by using Mathematica, thereby solving the problem that there isno calculation formula or method to calculate the natural circularfrequencies of different orders of the current multi-segment continuousbeams with different lengths, sizes and materials under given elasticboundary conditions.

Embodiment 1

Taking the cantilever multi-segmental continuous beam shown in thefigure as an example, after a mass matrix and a stiffness matrix of itsbending vibration are given, the natural circular frequency can becalculated through the matrix eigenvalue problem. This method issuitable for cantilever multi-segment beams with different segmentlengths, different cross-sectional shapes and different cross-sectionaldimensions.

As shown in the figure, a total length of the beam is L and the beam isdivided into 2 segments, where a length of a left segment is L₁; a massper unit volume of the left segment is ρ₁; an area of a cross section ofthe left segment is A₁; a moment of inertia of the cross section of theleft segment is I₁; and an elastic modulus of the left segment is E₁. Alength of a right segment is L₂; a mass per unit volume of the rightsegment is ρ₂; an area of a cross section of the right segment is A₂; amoment of inertia of a cross section of the right segment is I₂; and anelastic modulus of the right segment is E₂.

It is assumed that w(x, t) is the lateral displacement of the crosssection of the multi-segment continuous beam from the coordinate originx at the t moment.

Based on the variable separation method, the modal solution is set asfollows:

w(x,t)=W(x)e ^(iwt)   (4).

The function is set as follows:

${{Kp}_{1} = {E_{1}{I_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}},\ldots$ ${Kp_{p}} = {E_{p}{I_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}$ ${{Ks}_{1} = {k_{1}\begin{bmatrix}{f_{1}f_{1}} & {f_{1}f_{2}} & \ldots & {f_{1}f_{m}} \\{f_{1}f_{2}} & {f_{2}f_{2}} & \ldots & {f_{2}f_{m}} \\\vdots & \vdots & \vdots & \vdots \\{f_{1}f_{m}} & {f_{2}f_{m}} & \ldots & {f_{m}f_{n}}\end{bmatrix}}_{x = 0}},{{Ks}_{2} = {k_{2}\begin{bmatrix}{f_{1}f_{1}} & {f_{1}f_{2}} & \ldots & {f_{1}f_{m}} \\{f_{1}f_{2}} & {f_{2}f_{2}} & \ldots & {f_{2}f_{m}} \\\vdots & \vdots & \vdots & \vdots \\{f_{1}f_{m}} & {f_{2}f_{m}} & \ldots & {f_{m}f_{n}}\end{bmatrix}}_{x = L}},{{Ks}_{3} = {K_{1}\begin{bmatrix}{\frac{{df}_{1}}{dx}\frac{{df}_{1}}{dx}} & {\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} \\{\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} \\\vdots & \vdots & \vdots & \vdots \\{\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} & \ldots & {\frac{{df}_{m}}{dx}\frac{{df}_{m}}{dx}}\end{bmatrix}}_{x = 0}},{{Ks}_{4} = {K_{2}\begin{bmatrix}{\frac{{df}_{1}}{dx}\frac{{df}_{1}}{dx}} & {\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} \\{\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{2}}{dx}} & \ldots & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} \\\vdots & \vdots & \vdots & \vdots \\{\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} & \ldots & {\frac{{df}_{m}}{dx}\frac{{df}_{m}}{dx}}\end{bmatrix}}_{x = L}}$ ${M_{1} = {\rho_{1}{A_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{f_{1}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{1}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{1}f_{m}{dx}}} \\{\int_{0}^{L_{1}}{f_{2}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{2}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{2}f_{m}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{f_{m}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{m}f_{2}{dx}}} & \ldots & {\int_{0}^{L_{1}}{f_{m}f_{m}{dx}}}\end{bmatrix}}}},\ldots$ $M_{p} = {\rho_{p}{A_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{1}dx}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{1}f_{m}dx}} \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{1}{dx}}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{2}f_{m}dx}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{1}dx}} & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{2}dx}} & \ldots & {\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{f_{m}f_{m}dx}}\end{bmatrix}}}$

a matrix of the linear equation system is as follows:

[(M ₁ +M ₂)ω²−(Kp₁ +Kp ₂ +Ks ₁ +Ks ₂ +Ks ₃ +Ks ₄)]A=0   (8);

where A={a⁻⁴, a⁻³, . . . , a₈, a₉}^(T); based on the necessary andsufficient condition for the linear equations to have nontrivialsolutions, the determinant of the coefficients of the equations shouldbe zero to obtain the frequency equation:

|(M ₁ +M ₂)ω²−(Kp ₁ +Kp ₂ +Ks ₁ +Ks ₂ +Ks ₃ +Ks ₄)|=0   (9);

where ω is the circular frequency to be determined. The matrices M₁, M₂,Kp₁, Kp₂, Ks₁, Ks₂, Ks₃, and Ks₄ need to be established usingMathematica. The equation (9) corresponds to the eigenvalue problem ofthe matrix, where the eigenvalue problem of the matrix is verycomplicated, which cannot be solved for manually and can only be solvedfor by using Mathematica.

(1) The cross sections of the left segment and the right segment arecircular: it is assumed that in the figure, a diameter of the leftsegment is d₁; an area of the circular cross section of the left segmentis A₁=πd²/4; the axial moment of inertia of the left segment is I₁=πd₁⁴/64; a diameter of the right segment is d₂; an area of the circularcross section of the right segment is A₂=πd²/4; and the axial moment ofinertia of the right segment is I₂=π·d₂ ⁴/64.

A) When a ratio of L₁ to L₂ takes different values

A diameter of the circular cross section of the left segment is d₁=40mm; a diameter of the circular cross section of the right segment isd₂=30 mm; a total length of the beam is L=0.15 m, E₁=E₂=210 GPa,ρ₁=ρ₂=7800 kg/m³. Under the cantilever boundary condition, when theratio of the length L₁ of the left segment and the length L₂ of theright segment of the beam takes different values, the first-orderfrequency obtained by this method is compared with the result of thetraditional analytical method. As shown in Table 1, under the cantileverboundary condition, the first-order natural frequency of themulti-segment beam continuously increases as the ratio of the length L₁of the left segment to the length L₂ of the right segment decreases, anddecreases when it approaches 1.

TABLE 1 Natural frequencies (rad/s) of a double-segment beamcorresponding to the values of L₁/L₂ under C-F boundary L₁/L₂ ω 8 3.5 21.75 1.25 1 Analytical 8876.48 9556.34 10044.8 10125.6 10191.1 10090.8method This 8876.95 9537.11 10058.2 10167 10220.2 10152.4 method Error(%) 0 −0.20 0.13 0.41 0.29 0.61

B) When a ratio of d₁ to d₁ takes different values

A double-segment beam with a circular section under the cantileverboundary condition is selected, where a length of the left segment isL₁=0.117 m; a length of the right segment is L₂=0.033 m; E₁=E₂=210 GPa;and ρ₁=ρ₂=7800 kg/m³. Under the cantilever boundary condition, adiameter of the circular section of the left segment of the beam isd₁=0.04 m. After changing the ratio of the diameter d₁ of the circularsection of the left segment to the diameter d₂ of the circular sectionof the right segment of the beam, it can be found that the first-orderfrequency obtained by this method is consistent with the result of thetraditional analytical method through comparison. As shown in Table 2,under the cantilever boundary condition, the first-order naturalfrequency of the multi-segment beam continuously decreases as the ratioof the diameter d₁ of the circular section of the left segment to thediameter d₂ of the circular section of the right segment decreases.

TABLE 2 Natural frequencies of a double-segment beam corresponding tovalues of d₁/d₂ under C-F boundary d₁/d₂ ω 7 6 5 4 3 2 1 0.5 Analytical13048.9 12981.1 12864.7 12653.0 12222.5 11184.0 8108.31 4738.1 methodThis 13212.1 13090.2 12931.0 12652.1 12222.3 11182.0 8109.94 4769.04method Error (%) 1.3 0.84 0.52 0 0 −0.02 0.02 0.65

C) When the left segment and the right segment are of differentmaterials or different bending stiffness ratios

A double-segment beam with a circular section under the cantileverboundary condition is selected. The length of the left segment isL₁=0.117 m; the length of the right segment is L₂=0.033 m; ρ₁=ρ₂=7800kg/m³; a diameter of the circular section of the left segment is d1=0.04 m, and a diameter of the circular section of the right segment isd₂=0.038 m. Under the cantilever boundary condition, when the ratio ofE₁I₁ to E₂I₂ changes, the first-order frequency obtained by this methodis compared with the result of the traditional analytical method, andthe error is within the allowable range.

TABLE 3 Natural frequency (rad/s) of a double-segment beam correspondingto values of E₁I₁/E₂I₂ under C-F boundary Analytical This E₁ E₂ E₁I₁/method method Error (GPa) (GPa) E₂I₂ ω (rad/s) ω (rad/s) (%) 127 702.227 6508.14 6507.12 −0.02 206 120 2.108 8289.24 8289.86 0 108 68 1.9496002.42 5991.20 −0.19 145 103 1.729 6955.79 6947.16 −0.12 206 173 1.4628291.84 8307.24 0.19

(2) The cross sections of the left segment and the right segment arerectangular: it is assumed that a width of the cross section of the leftsegment L₁ in the figure is b₁; a height of the cross section of theleft segment L₁ is h₁; the area of the cross section of the crosssection of the left segment L₁ is A₁=b₁h₁; and the axial moment ofinertia of the cross section of the left segment L₁ is I₁=b₁h₁ ³/12; awidth of the cross section of the right segment L₂ is b₂; the height ofthe cross section of the right segment L₂ is h₂; the area of the crosssection of the right segment L₂ is A₂=b₂h₂; and the axial moment ofinertia of the cross section of the right segment L₂ is I₂=b₂h₂ ³/12.

A) When a ratio of L₁ to L₂ takes different values

The width b₁ of the left segment of rectangular segment is 40 mm, and aheight h₁ of the left segment of rectangular segment is 30 mm; the widthb₂ of the cross section of the right segment is 20 mm, and the height h₂of the cross section of the right segment is 15 mm; the total length Lof the double-segment beam is 0.15 m; E₁=E₂=210 GPa; ρ₁=ρ₂=7800 kg/m³.Under the cantilever boundary condition, when the ratio of the length L₁of the left segment to the length L₂ of the right segment takesdifferent values, it can be seen that the data of the first-orderfrequency obtained by this method is consistent with the result of thetraditional analytical method through comparison. As shown in Table 4,under the cantilever boundary condition, the first-order naturalfrequency of the multi-segment beam with rectangular cross sectionincreases with the decrease of the ratio of the length L₁ of the leftsegment to the length L₂ of the right segment, and decreases when itapproaches 1.

TABLE 4 Natural frequencies of a double-segment beam with rectangularsections corresponding to values of L₁/L₂ under the C-F boundary L₁/L₂ ω8 3.5 2 1.75 1.25 1 Analytical 8291.86 9714.37 10959.3 11137.3 10898.210125.5 method This 8323.36 9721.02 11062.0 11386.7 11481.0 10801.7method Error (%) 0.38 0.07 0.94 2.23 5.3 6.7

B) When a ratio of A₁ to A₂ takes different values

A double-segment beam with a rectangular cross-section under thecantilever boundary condition is selected, where the length of the leftsegment is L₁=0.117 m; a length of the right segment is L₂=0.033 m;E₁=E₂=210 GPa; and ρ₁=ρ₂=7800 kg/m³. Under the cantilever boundarycondition, the influence of the ratio of the area A₁ of the rectangularcross section of the left segment to the area A₂ of the rectangularcross section of the right segment on the first-order natural frequencyof the beam with the rectangular cross section is studied. It can beseen that the data of the first-order frequency is consistent with theresult of the traditional analytical method through comparison. As shownin Table 5, the first-order natural frequency of the multi-segment beamwith the rectangular cross section continuously decreases as the ratioof the area A₁ of the cross section of the left segment to the area A₂of the cross section of the right segment decreases under the cantileverboundary condition.

TABLE 5 Natural frequencies (rad/s) of a double-segment rectangular beamcorresponding to values of A₁/A₂ under C-F boundary Analytical This A₁A₂ A₁/ method method Error (mm²) (mm²) A₂ ω (rad/s) ω (rad/s) (%) 37 ×27 34 × 24 1.224 6724.62 6720.34 −0.06 45 × 35 42 × 32 1.172 8604.638598.65 −0.07 35 × 25 33 × 23 1.153 6115.92 6148.12 0.53 39 × 29 37 × 271.132 7055.65 7052.39 −0.05 40 × 30 38 × 28 1.128 7290.47 7297.26 0.09

C) When the left segment and the right segment are of differentmaterials or different bending stiffness ratios

A double-segment beam with a rectangular section under the cantileverboundary condition is selected. The length of the left segment isL₁=0.117 m, and the length of the right segment is L₂=0.033 m.ρ₁=ρ₂=7800 kg/m³, b₁×h₁=40 mm×30 mm, and b₂×h₂=20 mm×15 mm. As shown inTable 6, under the cantilever boundary condition, when the ratio of E₁I₁to E₂I₂ changes, the first-order frequency obtained by this method iscompared with the result of the traditional analytical method, and itcan be seen that the error is within the allowable range.

TABLE 6 Natural frequencies (rad/s) of a double-segment rectangular beamcorresponding to values of E₁I₁/ E₂I₂ under C-F boundary Analytical ThisE₁ E₂ E₁I₁/ method method Error (GPa) (GPa) E₂I₂ ω (rad/s) ω (rad/s) (%)127 70 29.03 7520.18 7551.17 0.41 206 120 27.47 9579.52 9609.41 0.31 10868 25.41 6937.95 6941.16 0.05 145 103 22.52 8041.87 8075.41 0.42 206 17319.05 9589.36 9605.10 0.16

The method of this embodiment is not limited to beams with specificboundaries and is applicable to beams with arbitrary elastic boundaries.At the same, it is applicable to both a single-segment beam and a beamwith multiple segments, which can provide excellent reference for theanalysis of the vibration characteristics of multi-segment continuousbeams in engineering applications. Thus, the method of the presentdisclosure has broad market prospects.

It should be understood that, the above-mentioned embodiments areillustrative of the present disclosure, but not intended to limit thepresent disclosure. Any modification and improvement made withoutdeparting from the spirit of the present disclosure shall fall withinthe scope of the invention which is defined by the appended claims andequivalents thereof.

What is claimed is:
 1. A method for calculation of a natural frequencyof a multi-segment continuous beam, comprising: (1) arranging adisplacement spring and a rotational spring on each of two ends of themulti-segment continuous beam to simulate arbitrary boundary conditions;(2) constructing a lateral displacement function of the multi-segmentcontinuous beam along a full length thereof, and expressing the lateraldisplacement function in a form of an improved Fourier series, whereinthe improved Fourier series is formed by adding four auxiliary functionsinto the classic Fourier series; (3) calculating a strain energy of themulti-segment continuous beam; (4) calculating an elastic potentialenergy of the displacement spring and the rotational spring at aboundary of the multi-segment continuous beam; (5) calculating a maximumvalue of a kinetic energy of the multi-segment continuous beam; (6)calculating a Lagrangian function of the multi-segment continuous beam;(7) substituting the improved Fourier series of the lateral displacementfunction into the Lagrange function; (8) taking an extreme value of eachof undetermined coefficients in the improved Fourier series in theLagrangian function to let a partial derivative be zero, so as to obtaina system of homogeneous linear equations; (9) converting the system ofhomogeneous linear equations into a matrix form; and (10) solving for aneigenvalue problem of the matrix to obtain the natural frequency.
 2. Themethod of claim 1, wherein in step (1), a stiffness value of thedisplacement spring and a stiffness value of the rotational spring at afirst boundary are respectively denoted as k₁ and K₁, and a stiffnessvalue of the displacement spring and a stiffness value of the rotationalspring at a second boundary are respectively denoted as k₂ and K₂; whenthe boundary is a clamped boundary, the stiffness value of thedisplacement spring and the stiffness value of the rotational springneed to be set to infinity at the same time, and the stiffness value ofthe displacement spring and the stiffness value of the rotational springare set to 10¹³, respectively; when the boundary is a free boundary, thestiffness value of the displacement spring and the stiffness value ofthe rotational spring are set to zero; when the boundary is a simplysupported boundary, the stiffness value of the displacement spring isset to 10¹³, and the stiffness value of the rotational spring is 0; andwhen the stiffness value of the displacement spring and the stiffnessvalue of the rotational spring are finite values, an elastic constraintboundary condition is simulated.
 3. The method of claim 1, wherein thelateral displacement function of the multi-segment continuous beam overthe full length thereof expressed in the form of the improved Fourierseries in step (2) is: $\begin{matrix}{{{W(x)} = {{\sum\limits_{n = 0}^{9}{a_{n}{\cos\left( {\lambda_{n}x} \right)}}} + {\sum\limits_{n = {- 4}}^{- 1}{a_{n}{\sin\left( {\lambda_{n}x} \right)}}}}};} & (1)\end{matrix}$ wherein x ∈[0,L]; a_(n) is an undetermined constant; andλ_(n)=nπ/L
 4. The method of claim 1, wherein the strain energy of themulti-segment continuous beam in step (3) is: $\begin{matrix}{{V_{P} = {{\frac{1}{2}E_{1}I_{1}{\int_{0}^{L_{1}}{\left( \frac{d^{2}w}{{dx}^{2}} \right)^{2}{dx}}}} + {\frac{1}{2}{\sum\limits_{i = 2}^{i = p}{E_{i}I_{i}{\int_{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i - 1}}}^{L_{1} + {L_{2}\mspace{11mu}\ldots\mspace{11mu} L_{i}}}{\left( \frac{d^{2}w}{dx^{2}} \right)^{2}{dx}}}}}}}};} & (2)\end{matrix}$ wherein a total length of the multi-segment continuousbeam is L; the multi-segment continuous beam is divided into p segments;a length of an i-th segment is L_(i); Vp is the strain energy of themulti-segment continuous beam under arbitrary boundary conditions; E_(i)is an elastic modulus of the i-th segment, and I_(i) is a moment ofinertia of a cross section of the i-th segment.
 5. The method of claim1, wherein the elastic potential energy Vs of the displacement springand the rotational spring at the boundary of the multi-segmentcontinuous beam in step (4) is: $\begin{matrix}{V_{s} = {\frac{1}{2}{\left( \left. {k_{1}w^{2}} \middle| {}_{x = 0}{+ {K_{1}\left( \frac{\partial w}{\partial x} \right)}^{2}} \middle| {}_{x = 0}{{+ k_{2}}w^{2}} \middle| {}_{x = L}{+ {K_{2}\left( \frac{\partial w}{\partial x} \right)}^{2}} \right|_{x = L} \right).}}} & (3)\end{matrix}$
 6. The method of claim 1, wherein a form of a modalsolution of the multi-segment continuous beam is assumed based on avariable separation method in step (2) as:w(x,t)=W(x)e ^(iwt)   (4); wherein i is an imaginary unit, and ω is thenatural frequency of the multi-segment continuous beam.
 7. The method ofclaim 1, wherein the maximum value of the kinetic energy of themulti-segment continuous beam in step (5) is: $\begin{matrix}{T_{{ma}\; x} = {{\frac{1}{2}{\rho(x)}{\int_{0}^{L}{{S(x)}\left( \frac{dw}{dt} \right)^{2}dx}}} = {\frac{\omega^{2}}{2}{\int_{0}^{L}{{\rho(x)}{S(x)}w^{2}{{dx}.}}}}}} & (5)\end{matrix}$
 8. The method of claim 1, wherein the Lagrangian functionof the multi-segment continuous beam in step (6) is: $\begin{matrix}{L = {{V_{{ma}\; x} - T_{m\;{ax}}} = {V_{p} + V_{s} - {T_{\;{{ma}\; x}}.}}}} & (6)\end{matrix}$
 9. The method of claim 1, wherein in step (8), the partialderivative of the undertermined coefficient an (n=−4, −3, . . . , 9) iscalculated item by item in the Lagrangian function, to obtain the systemof homogeneous linear equations:[(M ₁ + . . . +M _(p))ω²(Kp ₁+ . . . +Kp_(p)+Ks₁+Ks₂+Ks₃+Ks₄)]A=0   (7);wherein A={a⁻⁴, a⁻³, . . . , a₈, a₉}^(T),$\mspace{79mu}{{{Kp}_{1} = {E_{1}{I_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{0}^{L_{1}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}},\mspace{79mu}\ldots}$${Kp}_{p} = {E_{p}{I_{p}\begin{bmatrix}{\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{1}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\{\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{2}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}} \\\ldots & \ldots & \vdots & \ldots \\{\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{1}}{{dx}^{2}}{dx}}} & {\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{2}}{{dx}^{2}}{dx}}} & \ldots & {\int_{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\ldots\mspace{14mu} L_{i}}}{\frac{d^{2}f_{m}}{{dx}^{2}}\frac{d^{2}f_{m}}{{dx}^{2}}{dx}}}\end{bmatrix}}}$ $\mspace{79mu}{{{Ks}_{1} = \left. {k_{1}\begin{bmatrix}{f_{1}f_{1}} & {f_{1}f_{2}} & \cdots & {f_{1}f_{m}} \\{f_{1}f_{2}} & {f_{2}f_{2}} & \cdots & {f_{2}f_{m}} \\\vdots & \vdots & \vdots & \vdots \\{f_{1}f_{m}} & {f_{2}f_{m}} & \cdots & {f_{m}f_{m}}\end{bmatrix}} \right|_{x = 0}},\mspace{79mu}{{Ks}_{2} = \left. {k_{2}\begin{bmatrix}{f_{1}f_{1}} & {f_{1}f_{2}} & {\cdots\;} & {f_{1}f_{m}} \\{f_{1}f_{2}} & {f_{2}f_{2}} & \cdots & {f_{2}f_{m}} \\\vdots & \vdots & \vdots & \vdots \\{f_{1}f_{m}} & {f_{2}f_{m}} & \cdots & {f_{m}f_{m}}\end{bmatrix}} \right|_{x = L}},\mspace{79mu}{{Ks}_{3} = \left. {K_{1}\begin{bmatrix}{\frac{{df}_{1}}{dx}\frac{{df}_{1}}{dx}} & {\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & \cdots & {\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} \\{\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{2}}{dx}} & \cdots & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} \\\vdots & \vdots & \vdots & \vdots \\{\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} & \cdots & {\frac{{df}_{m}}{dx}\frac{{df}_{m}}{dx}}\end{bmatrix}} \right|_{x = 0}},\mspace{79mu}{{Ks}_{4} = {K_{2} = {\left. \begin{bmatrix}{\frac{{df}_{1}}{dx}\frac{{df}_{1}}{dx}} & {\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & \cdots & {\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} \\{\frac{{df}_{1}}{dx}\frac{{df}_{2}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{2}}{dx}} & \cdots & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} \\\vdots & \vdots & \vdots & \vdots \\{\frac{{df}_{1}}{dx}\frac{{df}_{m}}{dx}} & {\frac{{df}_{2}}{dx}\frac{{df}_{m}}{dx}} & \cdots & {\frac{{df}_{m}}{dx}\frac{{df}_{m}}{dx}}\end{bmatrix} \middle| {}_{x = L}\mspace{79mu} M_{1} \right. = {\rho_{1}{A_{1}\begin{bmatrix}{\int_{0}^{L_{1}}{f_{1}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{1}f_{2}{dx}}} & \cdots & {\int_{0}^{L_{1}}{f_{1}f_{m}{dx}}} \\{\int_{0}^{L_{1}}{f_{2}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{2}f_{2}{dx}}} & \cdots & {\int_{0}^{L_{1}}{f_{2}f_{m}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{0}^{L_{1}}{f_{m}f_{1}{dx}}} & {\int_{0}^{L_{1}}{f_{m}f_{2}{dx}}} & \cdots & {\int_{0}^{L_{1}}{f_{m}f_{m}{dx}}}\end{bmatrix}}}}}},\mspace{79mu}\ldots}$$\mspace{79mu}{M_{p} = {{\rho_{p}A_{p}} = {\begin{bmatrix}{\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{1}f_{1}{dx}}} & {\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{1}f_{2}{dx}}} & \cdots & {\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{1}f_{m}{dx}}} \\{\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{2}f_{1}{dx}}} & {\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{2}f_{2}{dx}}} & \cdots & {\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{2}f_{m}{dx}}} \\\vdots & \vdots & \vdots & \vdots \\{\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{m}f_{1}{dx}}} & {\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{m}f_{2}{dx}}} & \cdots & {\int_{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i - 1}}}^{L_{1} + {L_{2}\cdots\mspace{14mu} L_{i}}}{f_{m}f_{m}{dx}}}\end{bmatrix}.}}}$
 10. The method of claim 1, wherein a condition forthe system of the homogeneous linear equations to have a nontrivialsolution in the step (8) is: a value of coefficient determinant of thesystem of the homogeneous linear equations is zero to obtain a frequencyequation.